MathDB

5

Part of 2016 Brazil National Olympiad

Problems(1)

Reiterating a polynomial

Source: 38th Brazilian MO 2016 - Second Day, Problem 5

11/23/2016
Consider the second-degree polynomial P(x)=4x2+12x−3015P(x) = 4x^2+12x-3015. Define the sequence of polynomials P1(x)=P(x)2016P_1(x)=\frac{P(x)}{2016} and Pn+1(x)=P(Pn(x))2016P_{n+1}(x)=\frac{P(P_n(x))}{2016} for every integer n≥1n \geq 1.
[list='a'] [*]Show that exists a real number rr such that Pn(r)<0P_n(r) < 0 for every positive integer nn. [*]Find how many integers mm are such that Pn(m)<0P_n(m)<0 for infinite positive integers nn.
Brazilian Math Olympiad 2016algebrapolynomial